Fast inversion of Chebyshev-Vandermonde systems

Numerische Mathematik (Impact Factor: 1.61). 08/1995; 67(1). DOI: 10.1007/s002110050018
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This paper contains two fast algorithms for
inversion of
matrices of the first and second kind. They are
based on special representations of the Bezoutians of Chebyshev polynomials of
both kinds. The paper also contains the results of numerical
experiments which show that the algorithms proposed here are not only much
faster, but also more stable than other algorithms available. It is also
efficient to use the above two algorithms for solving Chebyshev--Vandermode
systems of equations with preprocessing.

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    • ", Tr [28], GO [16] P [25], Tr[28] BP[9] Chebychev-V Chebychev poly GO [14] GO [14] RO[26] Three-Term-V Real orthogonal poly Vs [29], GO [14] CR [10] Hi[27] Szegö–V Szegö polynomial O [23] O [24] BEGKO [1] Quasiseparable Quasiseparable BEGOT [4] BEGOT [6], BEGKO [2] Vandermonde polynomial BEGOT [5] BEGOTZ [7] Table 1: Fast O(n 2 ) inversion for polynomial–Vandermonde matrices. Inversion formulas and fast system solving are classical applications in Displacement Theory. "
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    ABSTRACT: The results on Vandermonde-like matrices were introduced as a generalization of polynomial Vandermonde matrices, and the displacement structure of these matrices was used to derive an inversion formula. In this paper we first present a fast Gaussian elimination algorithm for the polynomial Vandermonde-like matrices. Later we use the said algorithm to derive fast inversion algorithms for quasiseparable, semiseparable and well-free Vandermonde-like matrices having $\mathcal{O}(n^2)$ complexity. To do so we identify structures of displacement operators in terms of generators and the recurrence relations(2-term and 3-term) between the columns of the basis transformation matrices for quasiseparable, semiseparable and well-free polynomials. Finally we present an $\mathcal{O}(n^2)$ algorithm to compute the inversion of quasiseparable Vandermonde-like matrices.
    Full-text · Article · Jan 2014
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    • "Chebyshev–V. Chebyshev Gohberg-Olshevsky [12] Three–Term V. Real orthogonal Calvetti-Reichel [9] Szegö–Vandermonde Szegö Olshevsky [19] (H, m)-semiseparable– (H, m)-semiseparable BEGOTZ [4] Vandermonde (new derivation in this paper) "
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    ABSTRACT: We use the language of signal flow graph representation of digital filter structures to solve three purely mathematical problems, including fast inversion of certain polynomial-Vandermonde matrices, deriving an analogue of the Horner and Clenshaw rules for polynomial evaluation in a (H,m)-quasiseparable basis, and computation of eigenvectors of (H,m)-quasiseparable classes of matrices. While algebraic derivations are possible, using elementary operations (specifically, flow reversal) on signal flow graphs provides a unified derivation, reveals connections with systems theory, etc.
    Preview · Article · Apr 2010 · Linear Algebra and its Applications
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    • "Björck-Pereyra [2] Chebyshev-Vandermonde Chebyshev polynomials Gohberg-Olshevsky [14] Reichel-Opfer [26] Three-Term Vandermonde Real orthogonal polynomials Calvetti-Reichel [7] Higham [19] Szegö-Vandermonde Szegö polynomials Olshevsky [23] BEGKO [3] 1.2. "
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    ABSTRACT: Although Gaussian elimination uses O(n 3) operations to invert an arbitrary matrix, matrices with a special Vandermonde structure can be inverted in only O(n 2) operations by the fast Traub algorithm. The original version of Traub algorithm was numerically unstable although only a minor modification of it yields a high accuracy in practice. The Traub algorithm has been extended from Vandermonde matrices involving monomials to polynomial-Vandermonde matrices involving real orthogonal polynomials, and the Szegö polynomials. In this paper we consider a new more general class of polynomials that we suggest to call Hessenberg order m quasisseparable polynomials, or (H, m)-quasiseparable polynomials. The new class is wide enough to include all of the above important special cases, e.g., monomials, real orthogonal polynomials and the Szcgö polynomials, as well as new subclasses. We derive a fast O(n 2) Traub-like algorithm to invert the associated (H, m)-quasisseparable-Vandermonde matrices. The class of quasiseparable matrices is garnering a lot of attention recently; it has been found to be useful in designing a number fo fast algorithms. The derivation of our new Traub-like algorithm is also based on exploiting quasiseparable structure of the corresponding Hessenberg matrices. Preliminary numerical experiments are presented comparing the algorithm to standard structure ignoring methods. This paper extends our recent results in [6] from the (H,0)-and (H,1)-quasiseparable cases to the more general (H, m)-quasiseparable case.
    Full-text · Chapter · Dec 2009
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